On a fractional (p,q)-Laplacian equation with critical nonlinearities

In this paper, we consider the following fractional (p, q)-Laplacian equations with critical Hardy-Sobolev exponents {(-Delta)(p)(s)(1)u + (-Delta)(q)(s)(2)u = lambda|u|(r-2) + mu |u|(p)(s1)*(alpha)-2 u/|x|(alpha) in Omega, u = 0 in R-N\Omega, 0 < s(2) < s(1) < 1 < q < r <= p <N/s(1), lambda, mu > 0 are two parameters, 0 <= alpha < ps(1) and p(s1)* (alpha) = p(N-alpha)/N-ps(1) is the fractional Hardy-Sobolev critical exponent, Omega subset of R-N is an open bounded domain with smooth boundary. By using variational methods, we show that the problem has a nontrivial nonnegative weak solution.